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Solvent Effects on the Permeability of Membrane-Supported Gels Kristen L. Buehler† and John L. Anderson* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
The hydrodynamic (solvent) permeability of porous membranes filled with cross-linked polyacrylamide (PA) was measured as a function of pore size, surface chemistry, and solvent quality. The gel was formed by first soaking the membrane in the solution containing the acrylamide monomer, cross-linking agent, and initiator, and then polymerizing in situ using thermal activation. The water permeability of the gel-filled membranes at a fixed polymer volume fraction within the pores was insensitive to the pore diameter over the range 0.09-0.48 µm. There was no measurable effect of two different surface chemistries, indicating that the gel was trapped inside the porous matrix rather than covalently bonded to the internal membrane surfaces. The most important result was that there were large variations of the permeability when the quality of the solvent was made poorer by adding different alcohols to water. Going from a good solvent for PA (water) to poorer solvents (increasing alcohol concentration), the permeability increased. For example, the hydrodynamic permeability of the membrane-supported gels increased 100 times at alcohol concentrations for which bulk gel collapsed by a factor of 7 in volume. The large changes in permeability of the membrane-supported gels were reversible when the alcohol content of the solvent was reduced; the gel’s permeability in pure water was always recovered to within (15% after exposing the gel to alcohol-rich solutions. These results, along with results of partitioning and filtration experiments using colloidal particles, support the conjecture that a membrane-supported gel undergoes microsyneresis (coarsening of the microstructure) when exposed to a poor solvent but does not collapse as a bulk gel would. Introduction While gels have intrinsic properties that can be designed for molecular and colloidal separations, they suffer from a lack of physical robustness that precludes the application of significant physical and osmotic pressures. Without a physical support, a bulk gel can withstand only minimal pressure before it compresses and loses its desired microstructure. Likewise, solvent changes lead to swelling or collapse of an unsupported gel. To overcome these deficiencies, a “gel in a shell” concept1,2 has been developed; that is, a gel is confined within a semirigid microporous support. The support provides stability to the gel and prevents mechanical and osmotic damage over a broad range of operating conditions. Initial attempts to modify macroporous membranes to make them selective toward molecular species were based on fixing polymers to the pore wall by adsorption.3,4 These early attempts did not produce selective membranes mainly because the dimensions of the polymer molecules were insufficient to allow them to penetrate to the core region of the pore. Mika and Childs5 were able to fill a substantial portion of the pore with cationic polymer, such that changes in hydrodynamic permeability of 2-3 orders of magnitude were observed when the pH was varied. This result was interpreted in terms of contractions of the polymer chains toward the pore wall when the pH was increased above the isoelectric point of the polymer. Methods have been developed to form gels within porous supports such as chromatographic packing1,2 and membranes.6,7 Generally, these methods involve first † Current address: Institute for Defense Analyses, 1801 N. Beauregard St., Alexandria, VA 22311.
imbibing the porous matrix with the solution containing the monomer and cross-linking agent and then initiating the polymerization process, for example, through heating the matrix or exposing it to UV radiation. Kapur et al.6 found that membrane-supported polyacrylamide gels can withstand pressure gradients greater than 300 bar/cm (pressure difference of 3 bar over a membrane 100 µm thick) without any measurable effect on the permeability of the gel, and the diffusion of proteins 1-3 nm in size through these gel-filled membranes is greatly hindered even though the pores of the membrane are 100 times larger than the proteins.8 Furthermore, the water permeability of a negatively charged polyacrylamide gel supported in a porous membrane was found to be insensitive to the ionic strength of the solution, even though bulk (unsupported) ionic gels deswell when the ionic strength is increased.6 While we are also interested in possible effects of pore size and surface chemistry of the supporting membrane on the permeability of gels, this paper is mainly concerned with effects of solvent quality. When a good solvent is replaced by a poor solvent, the polymer chains want to collapse (deswell) to exclude the solvent, making the gel more dense.9 Such deswelling leads to a substantial increase in the volume fraction of polymer in the gel through the expulsion of solvent, which would make a bulk gel essentially impermeable to solvent at any feasible pressure gradient. However, when the gel is constrained within a rigid porous matrix, it is not clear what will happen to the hydrodynamic permeability because the gel might not deswell because of physical constraints imposed by the porous matrix. The collapse of unsupported gels when placed in poor solvents is conceptualized as the two steps10 illustrated in Figure 1. The first step is microsyneresis, the formation of spatially heterogeneous regions where some
10.1021/ie010321z CCC: $22.00 © 2002 American Chemical Society Published on Web 10/13/2001
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Figure 1. Microsyneresis (I) and macrosyneresis (II) during the deswelling of bulk gels when they are exposed to a poor solvent. The large dots represent regions of cross-linking. Macrosyneresis might be prohibited when the gel is formed within a microporous support.
domains become more dilute in polymer chains while other domains become very dense, the latter occurring where the local cross-link density is high. The second step (macrosyneresis) is a consolidation of the two domains into a more or less homogeneous bulk polymer after expulsion of the solvent. The first step is thought to be fast, and the second can take a long time. It is conceivable that a constrained gel, such as one formed in a porous matrix, allows the first step but not the second. If this were true, then the hydrodynamic permeability would greatly increase due to flow through the polymer-depleted domains. There are prior studies of the effects of solvent quality on the permeability of polymer networks and gels that bear on this work. For example, Mijnlieff and Jaspers11 measured the sedimentation rate of entangled networks of poly(R-methyl styrene) in cyclohexane and toluene. From these measurements one can determine the hydrodynamic permeability of the (un-cross-linked) network.12 The hydrodynamic permeability of the polymer network for cyclohexane at temperatures slightly above the Θ point was 3 times the value for toluene, which is a good solvent. Note that the solvent quality of cyclohexane was not so bad as to cause collapse and precipitation of the polymer, as would have occurred if the Θ temperature had been crossed. Mijnlieff and Jaspers ascribed the higher permeability in cyclohexane to fluctuations in local polymer density. Tokita13 measured the hydrodynamic permeability of cross-linked poly(N-isopropylacrylamide) in water as a function of temperature. This polymer has a lower critical point in water, so that the solvent quality decreases as temperature increases with a Θ point at 33.6 °C. The gel was chemically bound to the surfaces of the flow chamber so that it did not collapse under poor solvent conditions. From 5 to 30 °C there was only a modest increase (≈40%) in permeability; however, as the θ point was crossed, the permeability increased by almost 3 orders of magnitude. This result is consistent with a coarsening of the microstructure of the gel (step I in Figure 1) as the solvent quality was decreased. There are two objectives of this work. First, we looked for possible effects of membrane properties, specifically pore size and surface chemistry, on the hydrodynamic permeability of neutral polyacrylamide (PA) gels. The effective pore diameter, as determined from the permeability of the bare (no gel) membrane, varied from 0.09 to 0.48 µm, and two types of membranes with difference
surface chemistries were used. The second objective was to determine the effect of solvent quality on the permeability. This was done by mixing a good solvent (water) with nonsolvents (alcohols), creating a range of solvent quality for PA. The deswelling of bulk (unsupported) PA gels in alcohol/water mixtures was measured and used as a reference to interpret the permeability measurements for the supported gels. The partitioning and filtration of colloidal particles were also studied to provide more insight into the microstructure of the membrane-supported gels. The flow of liquids through gels is often modeled using hydrodynamic theory for flow through fixed arrays of impermeable fibers or spheres. In the next section, we discuss some of these theories. The experiments are then described and the results are presented. The data are fit to models for the gel based on either a random network of fibers or an array of spherical blobs, the latter perhaps being a better physical model for a gel that has experienced microsyneresis. The main result of this work is that the hydrodynamic permeability of the supported gels increased by 2 orders of magnitude under solvent conditions where the bulk gels collapsed by a factor of 7 in volume, and the changes in permeability were reversible as the solvent quality was first made poor (alcohol-rich) and then returned to a good condition (pure water). Hydrodynamic Permeability The hydrodynamic permeability (k) of a gel is related to its microstructure. It is defined for an isotropic porous medium by the Brinkman equation:
η η∇2v - ∇P - v ) 0, ∇‚v ) 0 k
(1)
where η is the solvent’s viscosity, v is the velocity field, and P is the pressure. At distances greater than about xk from no-slip boundaries (i.e., liquid/solid interfaces such as a pore wall), the Brinkman equation reduces to Darcy’s law:
k v ) - ∇P η
(2)
From the second part of eq 1, the pressure field is given by the conduction equation:
∇2P ) 0
(3)
Given boundary conditions on pressure, this equation is solved and the fluid velocity is then determined from eq 2. Models for k as a function of the microstructure of a fibrous medium have been reviewed by Kapur et al.6 For a homogeneous medium composed primarily of slender fibers of the same size, the permeability is expressed as
k ) af2f(φ)
(4)
where af is the radius of the fiber and φ is the volume fraction of the fibers (1- φ is the volume fraction of the solvent). The spatial arrangement and orientation of the fibers determines the function f(φ). Most theories for randomly distributed fibers give predictions for this function to within (20% of each other.6 The following expression is due to Jackson and James14 and is a best
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fit to published data over a six-decade range of af beginning at about 0.5 nm:
3 f(φ) ) [0.931 + ln(φ)] 20φ
(5)
Clague and Phillips15 extended previous theories for fibrous media by performing rigorous hydrodynamic calculations at the fiber level and allowing for two different fiber sizes (af). Recently Mika and Childs7 proposed a theory based on the correlation length of entangled polymer chains. Their model is that “blobs” of polymer chains produce the main obstacles to flow and the gel can be viewed as a system of rigid spheres around which the solvent must flow. The equivalent sphere diameter is taken as the correlation length (ξ) of the polymer network, and ξ is computed from scaling relations without taking into account effects of cross-linking. They used approximate expressions from the literature for flow around spherical obstacles to predict k as a function of polymer volume fraction and found reasonable agreement with experimental data by adjusting the Flory-Huggins interaction parameter for the polymer to reflect the quality of the solvent. By borrowing from the model of Mika and Childs7 that a cross-linked gel can be viewed as an array of spherical impermeable blobs, the hydrodynamic permeability of the system is expressed as
k)
2 2 a g(φ) 9φ s
(6)
where φ is the volume fraction of the blobs (and most of the polymer is in the blobs) and as is the blob radius. The function g(φ) accounts for hydrodynamic screening among the blobs and depends on the spatial arrangement of the blobs; it is determined by solving the Stokes equations for viscous flow at the length scale of the blob size. Note that g ) 1 when φ ) 0; in this case, each blob imparts a resistive force to the fluid given by Stokes’ law. The function g accounts for hydrodynamic screening among the blobs. The “cell model” was introduced by Happel16 who obtained the following expression:
g(φ) )
3 - 4.5φ1/3 + 4.5φ5/3 - 3φ2 3 + 2φ5/3
(7)
Kang and Sangani17 cite the calculations by Ladd18 for a random distribution of spheres; Ladd’s results are approximated over the range 0.01 < φ < 0.45 by the following empirical expression:
g(φ) = 0.7778 exp(-6.9φ)
(8)
This expression should be more accurate than Happel’s for disordered spheres, since Happel’s model implicitly assumes a regular array of spheres. The above two expressions are in quantitative agreement when φ > 0.2. Whether a gel is better modeled as a random fibrous media or a random array of fixed spherical blobs might depend on the solvent quality. In a good solvent, the polymer chains are more expanded and isolated from each other, so that the fibrous-media model might be more appropriate. However, in a poor solvent where microsyneresis occurs (see Figure 1), the array-ofspheres model might be more realistic. Knowing the
value of φ for the gel, the measured value of k can be used to compute the length parameter (af or as) of either model, but it cannot be used to determine which model is more appropriate for that system. This point is revisited in the Discussion section. Experiments Details of the experimental procedures are given by Buehler.19 The gels were made by polymerizing acrylamide with the cross-linker N,N′-methylene bisacrylamide using a thermally activated initiator AZAP (2,2′azobis(2-methylpropionamidine)dihydrochloride). The monomer and cross-linker densities in the reaction mixture were as follows for all experiments:
mass monomer mass cross-link agent ) 10%; ) 5% mass solvent mass monomer (9) These values were chosen because reaction mixtures with these concentrations yielded isosteric bulk (unsupported) gels; i.e., the polymer volume fraction of the gel in deionized water is about the same as that in the reaction mixture. Thin square sheets of bulk (unsupported) gels were prepared by injecting the gel-forming solution (monomer, cross-linker, and initatior) into Biorad casting molds. The molds were placed in plastic bags that were heat-sealed, and the bags were then placed into an oven at 85 °C for 90 min. Nitrogen was used to eliminate oxygen from the system, and the gel-forming solution was de-gassed just prior to placing the molds in the oven. After the gels had formed, they were stored at room temperature in deionized water which was replaced daily. Two types of porous support membranes were purchased from the Millipore Corporation (Bedford, MA)Durapore and MF-Millipore. Both were about 100 µm thick. Durapore membranes are porous polyvinylidene fluoride with a surface coating of a polyacrylate to make them hydrophilic. MF-Millipore membranes are made of mixed cellulose esters (nitrate and acetate) without a surface coating. Figure 2 shows an electron micrograph of each type of membrane.20 Some properties of the membranes are listed in Table 1. An effective pore diameter (deff) was determined from the measured hydrodynamic permeability of the membrane without polymer (kmo) and the Kozeny equation assuming cylindrical pores: 1/2 τ deff ) 32 kmo
[
]
(10)
where is the porosity of the membrane and τ is a tortuosity factor. The porosity was determined by weighing the dry membrane and then calculating from the mass and volume of the membrane and the density of the solid membrane matrix. τ has been experimentally determined to be 2.3 for Durapore membranes.6 The process to fabricate the membrane-supported gels is shown in Figure 3. The porous support membrane was sandwiched between two pieces of filter paper (Whatman, qualitative, P5, medium flow, and medium porosity). The purpose of the filter paper was to remove excess solution from the faces of the membrane; note that we obtained the best results with this Whatman filter paper. The sandwich (filter paper plus membrane) was soaked in the gel-forming solution at room temper-
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Figure 3. Method of polymerizing a cross-linked gel within the pores of the support membranes.
Figure 4. Apparatus to measure the volume flowrate (Q) of solvent versus applied pressure difference (∆P) across the membrane-supported gel.
Figure 2. Electron micrographs of the two types of support membrane (Millipore Corporation, Bedford, MA). Table 1. Properties of the Support Membranesa
membrane MF-Millipore MF-Millipore MF-Millipore Durapore Durapore
pore kmo diameterb thicknessb porosity (10-12 deffc (nominal, µm) (L, µm) () cm2) (µm) 0.05 0.10 0.22 0.10 0.22
105 125 150 105 125
0.69 0.66 0.79 0.70 0.67
0.71 0.09 1.47 0.13 24.6 0.48 4.55 0.22 9.53 0.32
a They were circular disks of radius 2.35 cm. b From the manufacturer. c See eq 10: τ ) 2.3.
ature under a nitrogen blanket after the solution had been purged with nitrogen and then degassed The filter paper was then removed, and the solution-filled membrane was placed in a nitrogen-purged plastic bag which was then heat-sealed. The bag containing the impregnated membrane was placed in the oven at 85 °C for 90 min to form the gel. After the gel had formed, the membrane was removed from the plastic bag and stored at room temperature in deionized water which was replaced daily. The mass of polymer within the membrane-supported gel was determined from the difference in weight between the membrane with gel, after the solvent had been evaporated, and the dry membrane before impregnation with the gel-forming solution. The volume fraction of polymer (φ) in the gel, based on the volume of the pores, was then calculated from the polymer mass, the specific volume of polyacrylamide (0.7 g/cm3),21 and the total pore volume of the membrane (determined from the porosity). While all gels were synthesized with
the same solution conditions as in eq 9 and the same process shown in Figure 3, there was a range of polymer volume fractions (0.04-0.10) in the gel within the pores. The mean value of φ was typically about 0.07, but the standard deviation was as high as 17% among the membranes of one type. The apparatus used to determine the hydrodynamic permeability of the membranes is shown in Figure 4. In the case of bare membranes (no gel), the permeability was so high that the flowrate was determined by collecting and weighing the effluent from the capillary. For the membranes with gel in the pores, the flow was much less and the flowrate through the membrane was measured by following the movement of a bubble in the glass capillary. Care was taken to ensure that the flowrate was constant (i.e., the velocity of the bubble was constant). Sample data for a bare membrane and a gel-filled membrane are shown in Figures 5 and 6. The superficial fluid velocity (vs) equals the flowrate of solvent (Q) divided by the membrane area (A ) 17.35 cm2). In all cases, the flowrate of solvent was proportional to the pressure difference, and there was no hysteresis between increasing and decreasing pressure, indicating that the gel was not deformed or damaged by the applied pressure. For one-dimensional flow through a membrane, eq 2 becomes
vs t
Q km ∆P ) A η L
(11)
where ∆P is the pressure drop across the membrane and L is the membrane thickness. Thus, the hydrodynamic permeability of the membrane (km) equals the slope of the lines drawn in Figures 5 and 6. The permeability of
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Figure 5. Solvent velocity (deionized water) versus pressure gradient across a bare (without gel in the pores) 0.22 µm (nominal) Durapore membrane. L is the membrane thickness, vs is the superficial velocity of the solvent (Q/A where A ) wetted membrane area), and η is the viscosity of the solvent. bar ) 105 N/m2. The slope of the straight line is the hydrodynamic permeability of the membrane (see eq 11).
tion of the particles was estimated using dynamic light scattering. Fitting a log-normal distribution of the particle diameter to the intensity autocorrelation function, we obtained a number-average particle diameter of 6.1 nm with a standard deviation of 1.2 nm for deionized water. This mean size is one-half the nominal value of 12 nm. The intensity-average diameter (〈d6〉/ 〈d5〉) was 17 nm. The suspensions remained stable in methanol/water mixtures. The filtration of the silica particles through gel-filled membranes was studied with an apparatus very similar to that shown in Figure 4. The particles were dispersed in deionized water or water/alcohol mixtures at a concentration of 1% by mass. Pressurized nitrogen was used to force the Ludox dispersion through the membrane. The permeate flowed through a spectrophotometer which measured the turbidity at near-0 from incidence, from which the silica concentration in the permeate (Cp) was determined. The sieving coefficient (χ) of the silica is the fraction that was convected through the membrane by the flow and is given by the ratio
χ)
Figure 6. Solvent velocity (deionized water) versus pressure gradient across a 0.22 µm (nominal) Durapore membrane with a cross-linked PA gel (φ ≈ 0.07) in the pores.
the gel itself (k) is obtained by factoring out the porosity and tortuosity of the porous membrane support:6
τ k ) km
(12)
For the porous membranes used here, we assume τ ) 2.3.6 The solvent quality was varied by mixing deionized water (DI) with methanol, ethanol, or 2-propanol. Water is a good solvent, and PA is essentially insoluble in the alcohols. When transferred from deionized water into a water/alcohol mixture, the bulk gels decreased in size. Since the amount of polymer in the gel was fixed while the solvent was able to move into or out of the gel, the ratio of polymer volume fractions in the gel between the two solvents is given by the inverse of the gel volumes (V):
VDI φmixture ) φDI Vmixture
(13)
φ was determined directly by weighing the gel, and then evaporating the solvent and weighing the polymer residue. By using the density of the solvent mixture and the specific volume of polyacrylamide (0.7 cm3/g), φ was calculated. To provide more insight into the changes in microstructure of the supported gel, we performed filtration and partitioning experiments with Ludox silica particles (DuPont) of nominal diameter 12 nm. The size distribu-
Cp Cf
(14)
where Cf is the concentration of silica in the well-mixed, high-pressure side of the cell. At the low solvent velocities in our experiments, there were no significant boundary layer effects on the upstream side of the membrane. χ was determined from measurements of Cp versus time, the initial value of Cf, and a material balance on the silica in the cell.19 The partitioning of the silica particles into supported gels was determined by both deionized water and 0.74 mole fraction methanol in water. This was done by first bathing the gel-filled membrane in the silica suspension for a sufficient time to equilibrate the particles between the gel and the suspension, and then measuring the change of silica concentration in the suspension outside the membrane. The partition coefficient (K) is defined as the ratio of the silica concentration in the gel (Cs(gel), based on the pore volume) to the concentration (Cs(soln)) in the solution bathing the membrane:
K)
Cs(gel) Cs(soln)
(15)
Cs(gel) was determined from a material balance based on the change in concentration of silica particles in the bathing solution. Results Role of Membrane Support. Figure 7 shows sample flowrate versus pressure for five different membranes without gel in the pores. The slopes yield the permeability kmo, from which the effective pore diameters (deff) in Table 1 were computed using eq 10. Note that the nominal pore diameter and the actual pore diameter do not match; this is not surprising since the pores do not have a capillary structure and the value of pore diameter depends on what method is used to determine it. There is about a 5-fold variation in pore size among the supports. Furthermore, the surface chemistry of the Durapore (polyacrylate coating) and the MF-Millipore (cellulose ester) are different.
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Figure 9. Deswelling ratio of bulk PA gels as a function of concentration and type of alcohol in water/alcohol mixtures. φDI is the polymer volume fraction in deionized water.
Figure 7. Solvent velocity (deionized water) versus pressure gradient across bare (without gel in the pores) membranes of different pore sizes and surface chemistries. See Table 1.
Figure 10. Solvent velocity versus pressure gradient across 0.22 µm (nominal) Durapore membranes with gel in the pores (φ ≈ 0.07).
Figure 8. Solvent velocity (deionized water) versus pressure gradient across membranes of different pore sizes and surface chemistries with gel in the pores. Table 2. Membrane Hydrodynamic Permeabilities for Deionized Water after the Gel Was Synthesized in the Poresa membrane
pore diameter (nominal, µm)
〈km〉 (10-15 cm2)
N
σk (% of mean)
MF-Millipore MF-Millipore MF-Millipore Durapore Durapore
0.05 0.10 0.22 0.10 0.22
3.3 2.4 3.0 2.4 3.0
8 6 6 8 14
29 24 24 18 22
a 〈k 〉 is the mean membrane permeability, N is the number of m gel-filled membranes studied, and σk is the experimental standard deviation of km among the N membrane.
Figure 8 shows sample data for five membranes now filled with the PA gel. Table 2 lists a summary of the data for the membrane permeabilities after gel was formed in the pores of N samples of each type of membrane. There are two important results. First, the membrane permeability was reduced by 3-4 orders of magnitude by the gel; and second, all the membranes had about the same permeability when gel was in the pores. The standard deviation of φ among the samples
can explain the difference in slope among the five membranes. The water permeability of the gel itself (k) was determined from the membrane permeability (km) using eq 12. Our values of k for over the range 0.04 < φ < 0.08 for the 0.22 µm (nominal) Durapore membranes are slightly lower than Tokita’s results13 for unsupported PA gels in water. A regression of our data fits eq 5 with af ) 0.45 nm, while the regression of Tokita’s data (for 2.2% cross-link density, versus 5% for our data) yields af ) 0.55 nm. Role of Solvent Quality. The ability of the alcohols to collapse the bulk gels is demonstrated in Figure 9. To assess the isotropy of deswelling, circular disks of diameter D were punched from the bulk gels in water. The diameter of the gel disk could be accurately measured as a function of the solvent mixture, but the thickness was difficult to measure. A log-log plot of φ versus D yielded a slope of -3.0 ( 0.1, thus demonstrating that the deswelling of the bulk gel was isotropic.19 The 0.22 µm (nominal) Durapore membranes were used as supports to study the effect of the three alcohols on gel permeability. Only those gel-filled membranes with km ) 2.00 ( 0.30 × 10-15 cm2 (k ) 6.9 × 10-15) for deionized water were used in these experiments; this permeability corresponds to φ ≈ 0.074 (see eq 5 with af ) 0.45 nm). Note that while reagent-grade methanol and 2-propanol were used, the ethanol contained some methanol and lacked the purity of the other two alcohols. While the presence of alcohol shrinks bulk gels, it coarsens the microstructure of supported gels as indicated by a large increase in the hydrodynamic permeability of the membrane. Figure 10 demonstrates this effect. The hydrodynamic permeability increased about
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Figure 11. Ratio of hydrodynamic permeability in an alcohol/ water mixture to the permeability in deionized water as a function of alcohol concentration across 0.22 µm (nominal) Durapore membranes. kDI ) 6.9 × 10-15 cm2.
Figure 13. Sieving coefficient (χ) of Ludox spheres as a function of methanol concentration.
Figure 12. Ratio of the hydrodynamic permeability in an alcohol/ water mixture to the permeability in deionized water versus the deswelling ratio for bulk gels.
50 times its value in switching from deionized water to a methanol/water mixture (0.46 mole fraction methanol). However, we could detect no difference between the alcohol content of the permeate compared to the feed. This result can be explained if most of the flow was through gaps in the gel formed during microsyneresis (see Figure 1) where the solvent composition would be more like the external solution, and little flow occurred through the regions of dense polymer where the solvent would be water-rich. A summary of the effect of the alcohols on the permeability is presented in Figure 11. Figure 12 plots the same permeability data versus the deswelling ratio of the bulk gels. Note that even the highest permeability (0.4 mole fraction 2-propanol) for the gel was only about 3% of the permeability of the bare membrane, so the gel still controlled the solvent flow. An important result is that the permeability of the gel was reversible to changes in both pressure and solvent composition. The reversibility with respect to changes in pressure drop (increasing versus decreasing pressure) is demonstrated in Figures 6 and 10; a bulk gel could not withstand these pressure gradients without compression and some destruction. Perhaps more surprising is the reversibility upon changes in solvent composition, as shown in Figure 11. The open symbols indicate the membrane-supported gel was exposed to ever increasing concentrations of alcohol, while the filled symbols indicate reduction of the alcohol mole fraction.
The original water permeability of the supported gel was always recovered to within (15% after the gel had been exposed to high concentrations of alcohol. This reversibility implies negligible damage to the supported gel under the osmotic stress of the alcohol solutions, even though the same solutions caused almost total collapse of the bulk gels. Filtration and Partitioning of Ludox Spheres. The supported PA gels were effective in filtering the silica particles as shown in Figure 13. The data appear erratic because of the expanded scale of χ; the experimental uncertainty was (0.05. These results are surprising for the case of high methanol mole fraction because the permeability is 50-100 times greater than that with deionized water. The partitioning experiments showed that the supported PA gel essentially excluded Ludox in the absence of methanol (K ) 0.01), but the microstructure opened up considerably at the high methanol concentration (K ) 0.82). The partitioning results are consistent with the increase in hydrodynamic permeability at the higher methanol concentration. Discussion Our results are qualitatively consistent with the experiments by Tokita.13 When the supported gel experienced a poor solvent condition, the hydrodynamic permeability increased by orders of magnitude. This indicates an opening of the microstructure of the gel, forming dilute regions of polymer that allow the solvent to flow without much hydrodynamic resistance. The fact that the low permeability was recovered when the alcohol content was reduced, even back to pure water, demonstrates that the gel was not irreversibly damaged when stressed by the poor solvent mixtures. One could argue that the increases in hydrodynamic permeability in alcohol/water mixtures might have been caused by a collapse of the gel within the pores of the membrane rather than by microsyneresis. To consider this possibility, we establish a model of a collapsing gel in a circular cylindrical pore of radius Ro ) deff/2 as shown in Figure 14. If the gel collapsed into a cylindrical plug of radius Ri, then the solvent would bypass the gel and flow only within the annulus defined by Ri < r
0.5. Our two data points are plotted as circles; the filled symbols assume the silica particle is 3 nm in radius (number average radius from light scattering), and the unfilled circles assume a ) 6 nm (nominal radius). We did not measure the value of hydrodynamic permeability for these membranes; it is assumed to be km ) 2.0 × 10-15 cm2 (k ) 6.9 × 10-15 cm2), which was typical for the membranes in this group. The datum point for the high methanol concentration (K ) 0.82) agrees with the theory when the number-average particle radius is used. However, we are not able to use this result to distinguish between the two geometric models (fibers versus spherical blobs) for the structure of the gel.
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Conclusions Rigid porous membranes provide an effective support for polymer gels and protect them against the disruptive forces of mechanical and osmotic pressure. When the pore diameter is large (90 nm or greater) relative to the hydrodynamic screening length of the polyacrylamide gel (xk ≈ 1 nm), there is no significant effect of pore size on the permeability of the supported gel (Figure 8 and Table 2). The surface chemistry of the support membrane also appears to be irrelevant as long as the membrane is hydrophilic. The hydrodynamic permeability of supported gels increases as the solvent quality for the gel decreases. Increases by a factor of 100 are possible without damaging the gel. Of the two models which would explain the permeability increase, microsyneresis (Figure 1) appears to be more likely based on our results for filtration and partitioning of silica particles, and because the collapsing gel (Figure 14) would have produced an increase in the permeability of a factor of 900, rather than the factor of 100 that was measured. Because the gels exhibited their original hydrodynamic permeability in pure water after exposure to high concentrations of alcohol, we conclude that membranesupported gels maintain their three-dimensional integrity even under extreme osmotic stresses. This robustness is an attribute that could be exploited to render supported gels useful in membrane technology. Acknowledgment We appreciate the financial support of the Chemical Engineering Department at Carnegie Mellon University and NSF Grant CTS-9122573. We thank BioSepra, Inc., for the use of their facilities, and S. B. Kessler and J. Y. Koo for their helpful advice on the synthesis procedure for forming the gels within the porous membranes. We also thank our colleague Gary Patterson for his important help in the light-scattering characterizations and for his insight into the microstructure of gels. J.L.A. is indebted to his mentor, John Quinn, for his teachings, guidance, and continuous support over the years. Literature Cited (1) Boschetti, E. Advanced sorbents for preparative protein separation purposes. J. Chromatogr. A 1994, 658, 207. (2) Boschetti, E.; Gurrier, L.; Girot, P.; Horvath, J. Preparative high-performance liquid chromatographic separation of proteins with HyperD ion-exchange supports. J. Chromatogr. B 1995, 664, 225. (3) Idol, W. K.; Anderson, J. L. Effects of adsorbed polyelectrolytes on convective flow and diffusion in porous membranes. J. Membr. Sci. 1986, 28, 269. (4) Kim, J. T.; Anderson, J. L. Diffusion and Flow through Polymer-Lined Micropores. Ind. Eng. Chem. Res. 1991, 29, 1008.
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Received for review April 11, 2001 Revised manuscript received August 2, 2001 Accepted August 8, 2001 IE010321Z
